Questions — AQA M1 (172 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA M1 2006 January Q3
3
  1. A small stone is dropped from a height of 25 metres above the ground.
    1. Find the time taken for the stone to reach the ground.
    2. Find the speed of the stone as it reaches the ground.
  2. A large package is dropped from the same height as the stone. Explain briefly why the time taken for the package to reach the ground is likely to be different from that for the stone.
    (2 marks)
AQA M1 2006 January Q4
4 Water flows in a constant direction at a constant speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A boat travels in the water at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water.
  1. The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(74 ^ { \circ }\) to the direction of motion of the water, as shown in the diagram. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c220e6c4-2676-4022-8301-7d720dc082b2-3_120_164_662_488} \captionsetup{labelformat=empty} \caption{Velocity of the water}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c220e6c4-2676-4022-8301-7d720dc082b2-3_126_186_667_890} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
    \end{figure}
    1. Find \(V\).
    2. Show that \(u = 3.44\), correct to three significant figures.
  2. The boat changes course so that it travels relative to the water at an angle of \(45 ^ { \circ }\) to the direction of motion of the water. The resultant velocity of the boat is now of magnitude \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the water is unchanged, as shown in the diagram below. $$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
    \includegraphics[max width=\textwidth, alt={}]{c220e6c4-2676-4022-8301-7d720dc082b2-3_132_273_1493_895}
    Velocity of the boat relative to the water
    \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-3_232_355_1498_1384} Find the value of \(v\).
    (4 marks)
AQA M1 2006 January Q5
5 A golf ball is projected from a point \(O\) with initial velocity \(V\) at an angle \(\alpha\) to the horizontal. The ball first hits the ground at a point \(A\) which is at the same horizontal level as \(O\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-4_227_602_484_735} It is given that \(V \cos \alpha = 6 u\) and \(V \sin \alpha = 2.5 u\).
  1. Show that the time taken for the ball to travel from \(O\) to \(A\) is \(\frac { 5 u } { g }\).
  2. Find, in terms of \(g\) and \(u\), the distance \(O A\).
  3. Find \(V\), in terms of \(u\).
  4. State, in terms of \(u\), the least speed of the ball during its flight from \(O\) to \(A\).
AQA M1 2006 January Q6
6 A van moves from rest on a straight horizontal road.
  1. In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    During the second stage, the van accelerates from \(4 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
    During the third stage, the van accelerates from \(12 \mathrm {~ms} ^ { - 1 }\) to \(16 \mathrm {~ms} ^ { - 1 }\).
    1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
    2. Find the total distance that the van travels during the 30 seconds.
    3. Find the average speed of the van during the 30 seconds.
    4. Find the greatest acceleration of the van during the 30 seconds.
  2. In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below.
    \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-5_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
    1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
    2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).
AQA M1 2006 January Q7
7 A builder ties two identical buckets, \(P\) and \(Q\), to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-6_360_296_502_904} The buckets are each of mass 0.6 kg .
    1. State the magnitude of the tension in the rope.
    2. State the magnitude and direction of the force exerted on the beam by the rope.
  2. The bucket \(Q\) is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket \(Q\) moves vertically downwards with the stone inside.
    1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
    2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.
AQA M1 2006 January Q8
8 A rough slope is inclined at an angle of \(25 ^ { \circ }\) to the horizontal. A box of weight 80 newtons is on the slope. A rope is attached to the box and is parallel to the slope. The tension in the rope is of magnitude \(T\) newtons. The diagram shows the slope, the box and the rope.
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-7_307_469_500_840}
  1. The box is held in equilibrium by the rope.
    1. Show that the normal reaction force between the box and the slope is 72.5 newtons, correct to three significant figures.
    2. The coefficient of friction between the box and the slope is 0.32 . Find the magnitude of the maximum value of the frictional force which can act on the box.
    3. Find the least possible tension in the rope to prevent the box from moving down the slope.
    4. Find the greatest possible tension in the rope.
    5. Show that the mass of the box is approximately 8.16 kg .
  2. The rope is now released and the box slides down the slope. Find the acceleration of the box.
AQA M1 2010 January Q1
1 Two particles, \(A\) and \(B\), are travelling in the same direction along a straight line on a smooth horizontal surface. Particle \(A\) has mass 3 kg and particle \(B\) has mass 7 kg . Particle \(A\) has a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and particle \(B\) has a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_186_835_653_593} Particle \(A\) and particle \(B\) collide and coalesce to form a single particle. Find the speed of this single particle after the collision.
AQA M1 2010 January Q2
2 A sprinter accelerates from rest at a constant rate for the first 10 metres of a 100 -metre race. He takes 2.5 seconds to run the first 10 metres.
  1. Find the acceleration of the sprinter during the first 2.5 seconds of the race.
  2. Show that the speed of the sprinter at the end of the first 2.5 seconds of the race is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. The sprinter completes the 100 -metre race, travelling the remaining 90 metres at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the total time taken for the sprinter to travel the 100 metres.
  4. Calculate the average speed of the sprinter during the 100 -metre race.
AQA M1 2010 January Q3
3 A particle of mass 3 kg is on a smooth slope inclined at \(60 ^ { \circ }\) to the horizontal. The particle is held at rest by a force of \(T\) newtons parallel to the slope, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_337_284_2023_879}
  1. Draw a diagram to show all the forces acting on the particle.
  2. Show that the magnitude of the normal reaction acting on the particle is 14.7 newtons.
  3. Find \(T\).
AQA M1 2010 January Q4
4 A ball is released from rest at a height of 15 metres above ground level.
  1. Find the speed of the ball when it hits the ground, assuming that no air resistance acts on the ball.
  2. In fact, air resistance does act on the ball. Assume that the air resistance force has a constant magnitude of 0.9 newtons. The ball has a mass of 0.5 kg .
    1. Draw a diagram to show the forces acting on the ball, including the magnitudes of the forces acting.
    2. Show that the acceleration of the ball is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    3. Find the speed at which the ball hits the ground.
    4. Explain why the assumption that the air resistance force is constant may not be valid.
AQA M1 2010 January Q5
5 The constant forces \(\mathbf { F } _ { 1 } = ( 8 \mathbf { i } + 12 \mathbf { j } )\) newtons and \(\mathbf { F } _ { 2 } = ( 4 \mathbf { i } - 4 \mathbf { j } )\) newtons act on a particle. No other forces act on the particle.
  1. Find the resultant force acting on the particle.
  2. Given that the mass of the particle is 4 kg , show that the acceleration of the particle is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  3. At time \(t\) seconds, the velocity of the particle is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\).
    1. When \(t = 20 , \mathbf { v } = 40 \mathbf { i } + 32 \mathbf { j }\). Show that \(\mathbf { v } = - 20 \mathbf { i } - 8 \mathbf { j }\) when \(t = 0\).
    2. Write down an expression for \(\mathbf { v }\) at time \(t\).
    3. Find the times when the speed of the particle is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
AQA M1 2010 January Q6
6 A small train at an amusement park consists of an engine and two carriages connected to each other by light horizontal rods, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-4_190_1038_420_493} The engine has mass 2000 kg and each carriage has mass 500 kg . The train moves along a straight horizontal track. A resistance force of magnitude 400 newtons acts on the engine, and resistance forces of magnitude 300 newtons act on each carriage. The train is accelerating at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Draw a diagram to show the horizontal forces acting on Carriage 2.
  2. Show that the magnitude of the force that the rod exerts on Carriage 2 is 550 newtons.
  3. Find the magnitude of the force that the rod attached to the engine exerts on Carriage 1.
  4. A forward driving force of magnitude \(P\) newtons acts on the engine. Find \(P\).
AQA M1 2010 January Q7
7 A ball is projected horizontally with speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) at a height of 5 metres above horizontal ground. When the ball has travelled a horizontal distance of 15 metres, it hits the ground.
\includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-4_433_1296_1674_338}
  1. Show that the time it takes for the ball to travel to the point where it hits the ground is 1.01 seconds, correct to three significant figures.
  2. Find \(V\).
  3. Find the speed of the ball when it hits the ground.
  4. Find the angle between the velocity of the ball and the horizontal when the ball hits the ground. Give your answer to the nearest degree.
  5. State two assumptions that you have made about the ball while it is moving.
AQA M1 2010 January Q8
8 A crate, of mass 200 kg , is initially at rest on a rough horizontal surface. A smooth ring is attached to the crate. A light inextensible rope is passed through the ring, and each end of the rope is attached to a tractor. The lower part of the rope is horizontal and the upper part is at an angle of \(20 ^ { \circ }\) to the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-5_344_1186_518_420} When the tractor moves forward, the crate accelerates at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The coefficient of friction between the crate and the surface is 0.4 . Assume that the tension, \(T\) newtons, is the same in both parts of the rope.
  1. Draw and label a diagram to show the forces acting on the crate.
  2. Express the normal reaction between the surface and the crate in terms of \(T\).
  3. Find \(T\).
AQA M1 2007 June Q1
1 A ball is released from rest at a height \(h\) metres above ground level. The ball hits the ground 1.5 seconds after it is released. Assume that the ball is a particle that does not experience any air resistance.
  1. Show that the speed of the ball is \(14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits the ground.
  2. Find \(h\).
  3. Find the distance that the ball has fallen when its speed is \(5 \mathrm {~ms} ^ { - 1 }\).
AQA M1 2007 June Q2
2 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface. Particle \(A\) has mass 2 kg and velocity \(\left[ \begin{array} { r } 3
- 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). Particle \(B\) has mass 3 kg and velocity \(\left[ \begin{array} { r } - 4
1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). The two particles collide, and they coalesce during the collision.
  1. Find the velocity of the combined particles after the collision.
  2. Find the speed of the combined particles after the collision.
AQA M1 2007 June Q3
3 A sign, of mass 2 kg , is suspended from the ceiling of a supermarket by two light strings. It hangs in equilibrium with each string making an angle of \(35 ^ { \circ }\) to the vertical, as shown in the diagram. Model the sign as a particle.
\includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-2_424_385_1790_824}
  1. By resolving forces horizontally, show that the tension is the same in each string.
  2. Find the tension in each string.
  3. If the tension in a string exceeds 40 N , the string will break. Find the mass of the heaviest sign that could be suspended as shown in the diagram.
AQA M1 2007 June Q4
4 A car, of mass 1200 kg , is connected by a tow rope to a truck, of mass 2800 kg . The truck tows the car in a straight line along a horizontal road. Assume that the tow rope is horizontal. A horizontal driving force of magnitude 3000 N acts on the truck. A horizontal resistance force of magnitude 800 N acts on the car. The car and truck accelerate at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-3_177_1002_580_513}
  1. Find the tension in the tow rope.
  2. Show that the magnitude of the horizontal resistance force acting on the truck is 600 N .
  3. In fact, the tow rope is not horizontal. Assume that the resistance forces and the driving force are unchanged. Is the tension in the tow rope greater or less than in part (a)? Explain why.
AQA M1 2007 June Q5
5 An aeroplane flies in air that is moving due east at a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the aeroplane relative to the air is \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north. The aeroplane actually travels on a bearing of \(030 ^ { \circ }\).
  1. Show that \(V = 86.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
  2. Find the magnitude of the resultant velocity of the aeroplane.
AQA M1 2007 June Q6
6 A box, of mass 3 kg , is placed on a slope inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The box slides down the slope. Assume that air resistance can be ignored.
  1. A simple model assumes that the slope is smooth.
    1. Draw a diagram to show the forces acting on the box.
    2. Show that the acceleration of the box is \(4.9 \mathrm {~ms} ^ { - 2 }\).
  2. A revised model assumes that the slope is rough. The box slides down the slope from rest, travelling 5 metres in 2 seconds.
    1. Show that the acceleration of the box is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    2. Find the magnitude of the friction force acting on the box.
    3. Find the coefficient of friction between the box and the slope.
    4. In reality, air resistance affects the motion of the box. Explain how its acceleration would change if you took this into account.
AQA M1 2007 June Q7
7 An arrow is fired from a point \(A\) with a velocity of \(25 \mathrm {~ms} ^ { - 1 }\), at an angle of \(40 ^ { \circ }\) above the horizontal. The arrow hits a target at the point \(B\) which is at the same level as the point \(A\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-4_195_1093_1594_511}
  1. State two assumptions that you should make in order to model the motion of the arrow.
    (2 marks)
  2. Show that the time that it takes for the arrow to travel from \(A\) to \(B\) is 3.28 seconds, correct to three significant figures.
  3. Find the distance between the points \(A\) and \(B\).
  4. State the magnitude and direction of the velocity of the arrow when it hits the target.
  5. Find the minimum speed of the arrow during its flight.
AQA M1 2007 June Q8
8 A boat is initially at the origin, heading due east at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then experiences a constant acceleration of \(( - 0.2 \mathbf { i } + 0.25 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. State the initial velocity of the boat as a vector.
  2. Find an expression for the velocity of the boat \(t\) seconds after it has started to accelerate.
  3. Find the value of \(t\) when the boat is travelling due north.
  4. Find the bearing of the boat from the origin when the boat is travelling due north.